The approximate frequency shift of a radar echo from a meteoroid is derived.
The origin of head echos observed during trail formation are discussed by
considering schematic models. The echo is either a reflection of the radar
signal from thermalized electrons in the meteoroid trail or from the plasma
electrons co-moving with the meteoroid itself. It is shown that in these
opposite models the frequency shifts of the echo relative to the radar emitter
frequency must be identical, even though the underlying physics differs.
Calculations were made that give the echo frequency as function of time,
allowing qualitative comparison with observations. Some typical observations
show a conspicuous absence of negative frequency shifts in the head echo in
forward scattering. Arguments are given that this can correspond to meteoroids
with the co-moving plasma in front.Comment: 10 pages, 9 figure

Wilschut, Hans W.

English

arXiv

The approximate frequency shift of a radar echo from a meteoroid is derived. The origin of head echos observed during trail formation are discussed by considering schematic models. The echo is either a reflection of the radar signal from thermalized electrons in the meteoroid trail or from the plasma electrons co-moving with the meteoroid itself. It is shown that in these opposite models the frequency shifts of the echo relative to the radar emitter frequency must be identical, even though the underlying physics differs. Calculations were made that give the echo frequency as function of time, allowing qualitative comparison with observations. Some typical observations show a conspicuous absence of negative frequency shifts in the head echo in forward scattering. Arguments are given that this can correspond to meteoroids with the co-moving plasma in front

Wilschut, Hans

ARTS repository - University of Groningen

   University of GroningenFrequency shifts of head echos in meteoroid trail formationWilschut, HansPublished in:WGN, Journal of the International Meteor OrganizationIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionEarly version, also known as pre-printPublication date:2020Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Wilschut, H. (in press). Frequency shifts of head echos in meteoroid trail formation. WGN, Journal of theInternational Meteor Organization, 48(4). https://arxiv.org/pdf/2006.16927.pdfCopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 11-09-2023Frequency shifts of head echosin meteoroid trail formationHans Wilschut∗July 1, 2020AbstractThe approximate frequency shift of a radar echo from a meteoroid isderived. The origin of head echos are discussed by considering schematicmodels.1 IntroductionMany aspects concerning radio echos from meteoroid trails can be understood interms of the line-oscillator model. The standard work for this is McKinley’s book[1], chapter 8 and 9. However, McKinley does not show how frequency shiftsnear the optimal reflection point can be calculated. In this paper the formalismof [1] is extended to include frequency shifts. In doing so a close relation appearsbetween the line-oscillator and what is referred to as the "moving ball" model.In the latter model frequency shifts occur because the radio echo will be Dopplershifted.Inspecting measured meteoroid radio echo’s such as shown in figures 2 and3 one can observe two distinct parts. The first part is a signal with a frequencydeviating from the transmitter frequency and the second part,which extendsover longer time, is centered around the transmitter frequency. This first part isoften referred to as the "head echo". Many authors interpret this signal in termsof Doppler shifts. The larger second part is well understood as a reflection fromthe trail left behind after a meteoroid passed. The free thermalized electrons inthe trail can cause strong reflections of the radio signal when their individualamplitudes add coherently. The head echo is produced during trail formation.In the line-oscillator description the scattering electrons, created as the trail isformed, will have relative phases such that it appears as a frequency shift awayfrom the emitter frequency. This is not the Doppler shift due to reflections on aco-moving plasma. The latter is assumed in the moving ball description. Somerecent work on Doppler shifts in head echos can be found in this journal. Seefor example [2, 3, 4].One aim of this paper is to derive an approximate value of the frequencyshift using the line-oscillator model of [1]. The notation of that work will befollowed unless otherwise indicated. For the convenience of the reader some of∗University of GroningenEmail: hwwilschut@gmail.comhome address: Sankt Augustin, Germany1arXiv:2006.16927v1  [astro-ph.EP]  30 Jun 2020the formalism in ref. [1] will be repeated here. The derivation of the frequencyshift in this model will be given in the next section. The third section considersthe case of Doppler shifts, showing the close connection with the line-oscillatormodel. In section 4 example calculations are made for a qualitative comparisonwith observations. Section 5 gives a suggestion why spectra like the one shownin fig. 2 with only a half head echo are seen more frequently than the one infig. 3 with a complete head echo. The final section contains some concludingremarks.2 Derivation of the frequency shiftWhen an ionized trail is made by a meteoroid (see fig. 1), the created freeelectrons act as individual scatterers, they re-emit the signal of a transmitter inall directions. These can be observed in a receiver contributing an amplitudedARdAR ∝ sin(2πft− 2π(R′1 +R′2)λ), (1)where λ is the transmitters wavelength and f its frequency. ()All other pa-rameters in this work are defined in fig. 1.) Each scattering contribution has adifferent phase depending on the distance (R′1 + R′2) the signal travels. Goingalong the trail the addition becomes coherent when d(R′1 +R′2)/dt = 0. At thispoint the path followed is a reflection, the specular condition. The length ofthis path is R1 +R2 and is the shortest path between transmitter and receivervia a point on the meteoroid trajectory. This point will be referred to as thespecular point. In fig. 1 also shows signal paths corresponding to back scatter-ing. This is the radar setup, where emitter and transmitter are at nearly thesame location. An arbitrary path has length 2R and the shortest path is 2R0.Near the specular point where s R0 one finds thatR′1 +R′2 ≈ R1 +R2 +s2 sin2 ω2(1R1 − s cosω+1R2 + s cosω)(2)≈ R1 +R2 +s2 sin2 ω2R1 +R2R1R2. (3)This reduces toR ≈ R0 +s22R0(4)for back scattering. Also note that in chapter 9 of [1] the notation was changed:s → f or f. Here we will use s consistently for the path of the meteoroid.Further note that, in general, ω 6= π/2 − φ because the trail may make anangle β with the plane where forward scattering takes places, in which casesin2 ω = 1− sin2 φ cos2 β.Central to the problem is the summation over individual scatterers alongthe trail. The model assumes a constant ionization density along the trail, theamplitude is then given byAR ∝∫ xx1sin(χ− πx22)dx . (5)2R1R2R1’R2’TX RXs𝜔𝜑𝜑R0Rt0Figure 1: [color on-line] TX and RX are the locations of the transmitter andreceiver, respectively. The signal path for back scattering (red) and for forwardscattering (blue) is shown for the optimal path corresponding to the reflection(specular) point, t0, (full lines) and the path of a contributing neighboring point(dashed lines) separated by a distance s.Figure 2: Typical example, of a strong head echo. The horizontal scale is onesecond long, the vertical scale 1.5 kHz. The slope corresponds to -9.7 kHz/sData were taken 05 May 2020 in Kampenhout (BE) using the 49990 kHz VVSbeacon near Ieper (BE)[5].3Following [1], we first evaluate the integral for back scattering where 2s =x(R0λ)1/2 and χ = 2πft + a time independent phase. Thus one sums from thebeginning of the trail until a point s(t), i.e. the length of the trail at time t.Without loss of generality x1 → −∞ can be assumed, as shown explicitly insection 3. One obtainsAR ∝(C(x) +12)sinχ−(S(x) +12)cosχ , (6)where C and S denote Fresnel integrals1. To see what this means for the fre-quency we look at times t < t0 and t > t0 avoiding the complex behavior neart0 by considering |x| ' 1. With this approximationA< ∝cos(χ− πx22 )πxand A> ∝ sin(χ)− cos(χ) +cos(χ− πx22 )πx. (7)As t approaches t0 the amplitude A< increases slowly compared with the fre-quency f . At each t the instantaneous frequency fi can be obtained by deter-mining the phase Φ of A<(t)Φ = χ− πx22= 2πft− 2π s2R0λ(8)and taking its derivative with respect to t, givingfi =12πdΦdt= f − 2s ds/dtR0λ. (9)Note that here s > 0 and ds/dt < 0, the shift is thus positive. In practice oneanalyses the change in the instantaneous frequency fi,dfidt= −2([dsdt]2+ sd2sdt2)1R0λ. (10)A simple model choice is a constant velocity where s = |V (t0 − t)|, so thatdfidt= −2V2R0λ. (11)To obtain the shift in forward scattering one simply replaces eq. 4 wit eq. 3 tofinddfidt≈ −V2 sin2 ωλR1 +R2R1R2. (12)For t > t0 there is no such simple derivation for the phase of A> possible. Inorder to get insight numerical calculations were done. These will be discussedin section 4.3 Derivation of the frequency shift in terms ofDoppler shiftsIn this section we consider the possibility that only a short part of the trailsurvives, so that it has a length x < 1. Thus where at the front free electrons1Here S(z) =∫ z0 sinπt22dt and C(z) =∫ z0 cosπt22dt4Figure 3: Less common head echo showing a shift that extends to negativeshifts. The slope is -4.9 kHz/s. Data taken on 09 June 2011 in Kampenhoutusing the 49970 kHz BRAMS beacon at Dourbes (BE)[5].are created they disappear at the back. Therefore, the trail has a constantlength ∆x. In fact for the observer it appears as a passing object, which maybeas well be the meteoroid itself. When this object passes point t0 it will give anecho. In this case one can approximate eq. 5 byAR ∝ sin(χ−πx22)∆x . (13)The phase of this amplitude is identical to that in eq. 8 and the same relationshold for the shifts. The shift continues for t > 0 where the instantaneousfrequency, fi, is lower than the emitter frequency f .The bistatic Doppler shift is given by − 1λd(R′1 +R′2)/dt. Using the sameapproximations as eq. 3ione arrives at the identical expression as in eq. 9. Thusthe shifts are the same as they must be, because of Galilean invariance. Amore informative evaluation will come from a numerical calculation in the nextsection.4 Example CalculationsAt this point it may be interesting to consider in more detail how the signalappears in observation. The result of section 2 and 3 can be generalized intoone expressionAR ∝ (C(x)− C(x−∆x)) sinχ− (S(x)− S(x−∆x)) cosχ , (14)where ∆x is the length of the object or trail as defined above. In the followingcalculation we use for s the backscatter configuration with V = 30 km/s, R0 =100 km, and an emitter frequency f = 50 MHz.First we consider the signal power, A2R(t), which is obtained by averagingover a time long with respect to the frequency but small with respect to x(t).In fig. 4 this quantity is shown, evaluated for various values of ∆x. For small5-5 0 50.00.51.01.5xsignalFigure 4: A2R(x(t)) for ∆x = 0.5, 1, 2, 4, 8,∞ (black, red, blue, green, cyan,magenta)Figure 5: [Color online] Time versus frequency of log(AR) obtained with a STFTprocedure. Left: passing trail or object. Right: Dynamic meteoroid trail. Seetext∆x the signal has a nearly Gaussian dependence, where it should be notedthat x = 1 corresponds to about one Fresnel zone (here it corresponds to t =√λR0/2V = 13 ms or a distance of 387 m). At this small distance the signalhas the characteristics of Fraunhofer slit scattering. For ∆x > 2 and x >1 one observes what are called Fresnel oscillations. ∆x = ∞ refers to thesituation discussed in section 2; the pattern seen here corresponds to Fresneledge scattering. The oscillation frequency in the Fresnel pattern (see chapter 8eq. (8-15) in [1]) is identical to the frequency shift calculated in section 2. Infact, historically, it has been a main tool for determining meteoroid velocitiesinstead of the frequency shift.To observe the frequency dependence of eq. 14, it should be displayed as atime dependent frequency spectrum similar to the observed data. This can bedone by a Short Time Fourier Transform (STFT) of the theoretical expressionsfor AR in eq. 14 (see appendix A). The results are shown in fig. 5. First wediscuss the spectrum (left) which is close to what was discussed in the previoussection. The size of the object or trail passing through the specular point is6taken to be ∆x = 1. The signal has the characteristic Doppler shift dependencewith a maximum at the specular point. Superimposed is a Fraunhofer diffractionspectrum. This pattern has to occur independent of the interpretation of theobject in terms of a a short-lived trail or as moving object. The spectrum onthe right is for ∆x =∞, the line oscillator model of section 2. It is dominatedby the ridge at fi − f = 0 at t > t0. Its origin is the stationary trail. Thediagonal in the spectrum has the same time-frequency dependence as the left-hand spectrum but without the diffraction pattern. This is the head echo fromthe trail as it is being formed.There are several observations to be made at this point: The Fresnel oscilla-tions found in fig. 4 are not seen in the spectrum. They appear by reducing theHann window in the STFT (see appendix A). Calculations were made for a win-dow width of 0.5, 1 and 2 Fresnel zones. For the short window the oscillationscan be seen, but at the cost of resolution in frequency. This is characteristicfor a Fourier transform where time and frequency resolution exclude each othermutually. In fig. 6 the frequency spectrum is shown at t = 0.2 s for the threeHann windows. The spectrum with the largest Hann window was also used toobtain fig. 5. The branch of the head echo is clearly seen for that window, whilefor the smallest window it has all but disappeared due to the reduced frequencyresolution. Notice that the head echo signal is more than 100 times smallercompared to the signal of the stationary trail. This indicates that measuringthe head echo at t > t0 requires that the receiver settings and the associatedanalysis programs need to take the time average in consideration. The observa-tion of the frequency shift at t < t0 is rather robust and thus also in an actualmeasurement. If possible, one could determine both the Fresnel oscillations andthe frequency shift by analyzing the data in two different ways by good timeand poor frequency resolution and vice versa, respectively. In this way one hastwo ways to measure the meteoroid velocity.One might expect measured events as in fig. 3 to be the most common.However, most observations are as in fig. 2. This is either due to problemsassociated with measuring the head exho at t > t0 for the reasons mentionedabove or because the line-scattering model is not describing the meteoroid eventsadequately. Another reason for this can be the observational bias for strongecho’s with a short trail below the specular point.5 The half-ball modelThe characteristic asymmetric behavior of the head echo noticed at the endof the previous section can be resolved assuming reflections from the plasmain front of – and moving with – the meteoroid. Research around 2000, forexample with Arecibo [6] and ALTAIR [7] shows the importance of the headecho plasma. Such a plasma reflects radio waves like a metal mirror. Thespecular condition is then irrelevant since there will always be a spot on a sphereallowing reflection from transmitter to receiver; also in forward scattering. Butin forward scattering it is important to realize that only the front half of themeteoroid has this property. The situation is sketched in fig. 7. Only in theapproaching phase reflections are possible up to the specular point, after thatreflections would have to come from the back of the meteoroid. Therefore, inthis scenario a true Doppler shift signal is observed until t = t0, and after7-1000 -500 0 500 1000-15-10-50frequency fi-f [Hz]Log[�]Figure 6: [Color online] Frequency spectrum of log(AR) at t = 0.2 s the green,orange and blue curves refer to an average over 0.5, 1, and 2 Fresnel zones,respectively. The branch of the head echo is at -700 Hz. See also textthat only the trail left by the meteoroid at the specular point contributes tothe signal. This scenario, which could be called the "half-ball model", wouldexplain experimental data as in fig. 2. Reflections at the specular point proceedvia the shortest path between transmitter and receiver and therefore also givethe strongest reflection in this model, as it does in the line-oscillator model.It will be interesting to combine line-oscillator model (observing the back tail)with the "half-ball model" as the relevant trail parts do not pass the specularpoint at exactly the same time. In any case it is somewhat surprising that thehead plasma reflection is strong enough to be seen in a comparatively modestamateur setup.6 ConclusionsTwo very different approaches lead to the same shifts in the radio-echo frequencywhen a meteoroid passes through a point where the specular condition is fulfilled.One assumes either thermalized electrons in the local atmosphere or electronsco-moving with the meteoroid as the scatterers. They lead to the same frequencyshifts because they are both based on the time derivative of eq. 3. The modeldiscussed in section 2 describes head echos and the resulting stationary trailin a unified way. More explicitly, further calculations in section 4 show thatfrequency shift and the Fresnel diffraction pattern relates to the same observableeq. 9. In section 3 it was shown that a short-lived trail and a moving object givethe same frequency shift of the head echo as in section 2. The size of the objectmust have |x| / 1, i.e one Fresnel zone. The calculations show that in these casesa diffraction pattern should be visible on the head echo. In practice this is notobserved. This may be because of the simplicity of the model. The calculationsin section 4 allow for a qualitative comparison with actual observations. Thecontradicting requirements for frequency and time resolution were pointed outand it was also shown that this is reflected in the duality observing either the8R1R2R1’R2’TX RXt0t < t0 t = t0 t > t0Figure 7: [Color online] Half-ball model. An echo can be seen at any point alongthe meteoroid trajectory. But only from the front plasma of the meteoroid. Afterpassing the specular point (t = t0), there will be no echo any longer except fromthe stationary trail. The right hand side shows a calculation where this wasimplemented in the line-oscillator model although this is not the right modelfor such a scenario.Fresnel diffraction or the frequency shifts. This duality may also play a rolein the absence of a head echo at t > 0 in most observations. Another reasonfor this can be the observational bias for strong echo’s with a short trail belowthe specular point. However, assuming that a co-moving plasma in front of themeteoroid is important, there is a simple geometric argument why there is nohead echo after the meteoroid passes the specular point t0.As a final remark and recommendation: The measurements of meteoroidechos contain much more information than the frequency shifts. In particularthe rapid decay of the signal after the head echo, occurring within a second, maycontain more hints as to the precise nature of the meteoroid event. Modernequipment allows a quantitative measurement of the signal strength and theclassic text of [1] provides several methods for its analysis.A Appendix STFTThe Short Time Fourier Transform has here the following formAR(fi − f, t) =∫ ∞−∞AR((2πf(t+ t′), x(t+ t′))ei2π(fi−f)(t+t′)HannWindow(αt′)dt′ ,HannWindow(w) =12 +12 cos(2πw) −12 ≤ w ≤120 |w| > 12,where α determines the width of the window.9AcknowledgementThe author thanks F. Verbelen, W. Kaufmann, and M.T. German for clarifyingsome of the concepts in radio meteoroid measurements, for sharing their data,and for discussions.References[1] D.W.R. McKinley, "Meteor Science and Engineering", McGraw-Hill NewYork,1961.[2] W. Kaufmann, WGN, JIMO 48 (1998)12-16[3] F. Verbelen, WGN,JIMO 47 (2019)49-54[4] M.T. German, WGN, JIMO 48(2020)4-11[5] F. Verbelen, private communication[6] J.D. Mathews et al., Icarus 126(1997)157-169[7] R. M. Suggs et al., NASA report SEE/TP-2004-40010

Frequency shifts of head echos in meteoroid trail formation

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